| L(s) = 1 | − 2.02·2-s + 2.10·4-s + 3.48·5-s + 1.37·7-s − 0.213·8-s − 7.05·10-s + 0.191·11-s + 4.24·13-s − 2.77·14-s − 3.77·16-s + 4.07·17-s − 5.16·19-s + 7.32·20-s − 0.388·22-s + 0.978·23-s + 7.11·25-s − 8.59·26-s + 2.88·28-s + 10.0·29-s − 4.04·31-s + 8.08·32-s − 8.26·34-s + 4.76·35-s + 2.15·37-s + 10.4·38-s − 0.744·40-s + 0.0317·41-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.05·4-s + 1.55·5-s + 0.517·7-s − 0.0755·8-s − 2.23·10-s + 0.0577·11-s + 1.17·13-s − 0.741·14-s − 0.944·16-s + 0.989·17-s − 1.18·19-s + 1.63·20-s − 0.0827·22-s + 0.204·23-s + 1.42·25-s − 1.68·26-s + 0.545·28-s + 1.85·29-s − 0.727·31-s + 1.42·32-s − 1.41·34-s + 0.806·35-s + 0.353·37-s + 1.69·38-s − 0.117·40-s + 0.00496·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375853216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375853216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 0.191T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 - 0.978T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 0.0317T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 0.680T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.22T + 83T^{2} \) |
| 89 | \( 1 + 9.94T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.912774455415244482367454501773, −8.622243647072224899491188177203, −7.81147189441310216630627097682, −6.77729511377009609566109783554, −6.15664039348435529420007979032, −5.33289842055830907454995179795, −4.24904153264549455566694763481, −2.74790510535910751762534630128, −1.75565091585114491665158062310, −1.06305757238960019216776506424,
1.06305757238960019216776506424, 1.75565091585114491665158062310, 2.74790510535910751762534630128, 4.24904153264549455566694763481, 5.33289842055830907454995179795, 6.15664039348435529420007979032, 6.77729511377009609566109783554, 7.81147189441310216630627097682, 8.622243647072224899491188177203, 8.912774455415244482367454501773
